A futon or Japanese-type mattress which can be converted to use as a sofa or mattress, is a flexible piece of furniture that can be placed on the floor to be slept on, or placed on a frame to sit on, or used in other ways. In order to provide considerable sleeping comfort, the mattress must have a considerable thickness and provide a substantially even support surface. Such mattresses with a substantial thickness and a substantially even upper support surface, are very bulky and therefore difficult to store and transport. Typical sizes used in the United States are referred to as twin (39".times.75"), full (54".times.75"), and queen (50".times.80"). A mattress with a substantially even upper support surface and considerable thickness to provide comfortable support for sleeping and sitting, which could be compactly folded for storage and shipment, would be of considerable value. A method for further compacting the folded mattress would also be of considerable value.